'The circuit that Michael has shown (and slight variations) have been in use (in welding sets by Migatronic(Danish) and ESAB (Swedish) for at least 35 years or more..
It works , and works reliabably..'

So, they also use wall wart power supplies?
Well, since 1996 CE standards have to be fullfilled and I assume that those 'slight
variations' just handle this fact.
Wall wart power supply is sold for household applications and so, consequently protection class II is used. This class means protection is achieved by a double isolation from mains. Such power supplies have no connection to earth connector (green-yellow wire) and they are not intended to become connected to earth wire. More, power supply itself is even not allowed to have such connection! But many users just make this connection, e.g. when connecting a PC, or other stuff having some earth connection. Protection class I demands such earth connection at this equipement, for security purpose.
In all aplications where such combination of class II and class I products is commonly used or cannot be excluded, power supply is in danger. Have a look inside cd-players, recorders and such stuff of entertainement electronics, where this connection practise is widely used, because having more earthed than antenna of reciever would produce ground loop. In almost all of these devices, even the cheapest, rectifier diodes are decoupled by capacitors. Sometimes, when having external power supply even cables with wide band rejection chokes
are used, which has a similar effect like those decoupling capacitors.

Have a look at a real protection class II aplication: Battery charger. There is no connection to any earth, or earthed equipement. Everything is isolated from user, according to double isolation feature demanded by this norm. So, when there is a transient on mains, high voltage has absolutely no chance to damage anything inside. Why? Because extremely small stray capacitance to earth of only a fraction of picofarad builds a 'natural' barrier for high voltage. Or, speaking in terms of electronics: Very small stray capacitance forms a voltage divider in combination with junction capacitance, resulting in absolutely harmless voltages at sensible parts.
That's the way a wall wart sold power supply is intended to be used! But beware of any connection to earth. Then you would need some protection for rectifier diodes.

Bye,
Kai